Exercise
May 13'23
Answer
Key: B
[[math]]
\begin{aligned}
L(q) &= \left [ \binom{2}{0}(1-q)^2\right]^{5000} \left[ \binom{2}{1}q(1-q)\right]^{5000} = 2^{5000}q^{5000}(1-q)^{15000} \\
l(q) &= 5000 \ln(2) + 5000 \ln(q) + 15000 \ln(1 − q) \\
l^{'}(q) &= 5000q^{−1} − 15000(1 − q)^{−1} = 0 \\
\hat{q} &= 0.25 \\
l(0.25) &= 5000 \ln(2) + 5000 \ln(0.25) + 15000 \ln(0.75) \\
&= −7780.97.
\end{aligned}
[[/math]]
Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.